On the radicality property for spaces of symbols of bounded Volterra operators

Abstract

In a recent paper of the authors together with A. Aleman, it is shown that the Bloch space B in the unit disc has the following radicality property: if an analytic function g satisfies that g(n )is an element of B, then g(m) is an element of B, for all m <= n. Since B coincides with the space T(A(alpha)(p)) of analytic symbols g such that the Volterra-type operator T(g)f(z)=integral(z)(0)f(zeta)g '(zeta)d zeta is bounded on the classical weighted Bergman space A(alpha)(p), the radicality property was used to study the composition of paraproducts T-g and S(g)f=T(f)g on A(alpha)(p). Motivated by this fact, we prove that T(A(omega)(p)) also has the radicality property, for any radial weight omega. Unlike the classical case, the lack of a precise description of T(A(omega)(p)) for a general radial weight, induces us to prove the radicality property for Ap omega from precise norm-operator results for compositions of analytic paraproducts.